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Mathematical Foundations of Statistical Mechanics


Mathematical Foundations of Statistical Mechanics Cover


Synopses & Reviews

Publisher Comments:

The translation of this important book brings to the English-speaking mathematician and mathematical physicist a thoroughly up-to-date introduction to statistical mechanics.

It offers a precise and mathematically rigorous formulation of the problems of statistical mechanics, as opposed to the non-rigorous discussion presented in most other works. It provides analytical tools needed to replace many of the cumbersome concepts and devices commonly used for establishing basic formulae, and it furnishes the mathematician with a logical step-by-step introduction, which will enable him to master the elements of statistical mechanics in the shortest possible time.

After a historical sketch, the author discusses the geometry and kinematics of the phase space, with the theorems of Liouville and Birkhoff; the ergodic problem (in the sense of replacing time averages by phase averages); the theory of probability; central limit theorem; ideal monatomic gas; foundation of thermodynamics, and dispersion and distribution of sum functions.

"An excellent introduction to the difficult and important discipline of Statistical Mechanics. It is clear, concise, and rigorous. There is a very good chapter on the ergodic theorem (with a complete proof!) and . . . a highly lucid chapter on statistical foundations of thermodynamics . . . useful to teachers . . . and to mathematicians." ― M. Kac, Quarterly of Applied Mathematics.


Phase space, ergodic problems, central limit theorem, dispersion and distribution of sum functions. Chapters include Geometry and Kinematics of the Phase Space; Reduction to the Problem of the Theory of Probability; and more.

Table of Contents


Chapter I. Introduction

1. A brief historical sketch

2. Methodological characterization

Chapter II. Geometry and Kinematics of the Phase Space

3. The phase space of a mechanical system

4. Theorem of Liouville

5. Theorem of Birkhoff

6. Case of metric indecomposability

7. Structure functions

8. Components of mechanical systems

Chapter III. Ergodic Problem

9. Interpretation of physical quantities in statistical mechanics

10. Fixed and free integrals

11. Brief historical sketch

12. On metric indecomposability of reduced manifolds

13. The possibility of a formulation without the use of metric indecomposability

Chapter IV. Reduction to the Problem of the Theory of Probability

14. Fundamental distribution law

15. The distribution law of a component and its energy

16. Generating functions

17. Conjugate distribute functions

18. Systems consisting of a large number of components

Chapter V. Application of the Central Limit Theorem

19. Approximate expressions of structure functions

20. The small component and its energy. Boltzmann's law

21. Mean values of the sum functions

22. Energy distribution law of the large component

23. Example of monatomic ideal gas

24. The theorem of equipartition of energy

25. A system in thermal equilibrium. Canonical distribution of Gibbs

Chapter VI. Ideal Monatomic Gas

26. Velocity distribution. Maxwell's law

27. The gas pressure

28. Physical interpretation of the parameter

29. Gas pressure in an arbitrary field of force

Chapter VII. The Foundation of Thermodynamics

30. External parameters and the mean values of external forces

31. The volume of the gas as an external parameter

32. The second law of thermodynamics

33. The properties of entropy

34. Other thermodynamical functions

Chapter VIII. Dispersion and the Distributions of Sum Functions

35. The intermolecular correlation

36. Dispersion and distribution laws of the sum functions


  The proof of the central limit theorem of the theory of probability



Product Details

Khinchin, Alexander I.
Dover Publications
Gamow, George
Gamow, George
Khinchin, A. Ya.
Khinchin, A. Ya
New York
Mathematical Physics
Probability & Statistics - General
Mechanics - General
Edition Description:
Trade Paper
Dover Books on Mathematics
Publication Date:
December 1949
8.5 x 5.38 in 0.5 lb

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Mathematical Foundations of Statistical Mechanics New Trade Paper
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Product details 206 pages Dover Publications - English 9780486601472 Reviews:
"Synopsis" by ,
Phase space, ergodic problems, central limit theorem, dispersion and distribution of sum functions. Chapters include Geometry and Kinematics of the Phase Space; Reduction to the Problem of the Theory of Probability; and more.
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